The most common appeal to bias is an appeal to motive (or circumstantial ad hominem), in which it is asserted that the opponent's argument is false because they have some (base) motive for advancing it and are thus biased against the truth.

X's bias is not a determinant of Y's truth. If X is the most disgustingly biased individual in existence, but then calls the sky blue, does X's bias change the color of the sky? Rejecting X as closeminded does not entail anything else.

All shill gambits are appeals to bias. In many cases there is no actual evidence of the supposed motive. A common example is the Big Pharma Conspiracy: defenders of woo will dismiss those advancing rational argument as being "big pharma shills" or part of "$cience":

“”The determination to 'discredit' Burzynski must be money-driven, like the wealthy ACS

Appeal to motive is a logical fallacy. However, motive can be highly relevant in practical discourse: when almost all the proponents of position A seem to support rather more odious position B, then it's a useful heuristic to check out whether B is actually the goal. e.g.Holocaust denial is, in practice, overwhelmingly just neo-Nazism dressed up as a mere question of history. Similarly, much of the denial of the health effects of tobacco was funded by tobacco companies.

If one can prove the opponent's argument is false, and that they have an ulterior motive for making the false claim, their entire position is much weaker. But to argue robustly, you need to show the actual argument is wrong. Simply being indicated as having a conflict of interest does not make one wrong.

It is also helpful as a crutch when assessing sources when pressed for time. Take, for example, Julius Caesar.

Caesar is obviously a biased source when describing his own exploits in Gaul, so when he says something that makes Julius Caesar look bad — say, describing a battle where he got shellacked — it is rather more likely to be true. Similarly, a noted detractor of Caesar writing something that makes Caesar look good is also more likely to be true. After all - what reason would they have to lie if it goes against their obvious biases? However, as mentioned earlier, it is only a crutch and people can be lying or mistaken even in public written statements that make them, their cause or their side look bad.